cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A034956 Divide natural numbers in groups with prime(n) elements and add together.

Original entry on oeis.org

3, 12, 40, 98, 253, 455, 850, 1292, 2047, 3335, 4495, 6623, 8938, 11180, 14335, 18815, 24249, 28731, 35845, 42884, 49348, 59408, 69139, 81791, 98164, 112211, 124939, 141026, 155434, 173681, 210439, 233966, 263040, 286062, 328098, 355152, 393442, 434558, 472777
Offset: 1

Views

Author

Patrick De Geest, Oct 15 1998

Keywords

Comments

Natural numbers starting from 1,2,3,4,...

Examples

			{1,2} #2 S=3;
{3,4,5} #3 S=12;
{6,7,8,9,10} #5 S=40;
{11,12,13,14,15,16,17} #7 S=98.

Links

Crossrefs

Cf. A006003, A027441, A034957.
Cf. A046992, A034958, A034959, A034960.
Cf. A000040, A000217, A007504.

Programs

  • Maple
    s:= proc(n) s(n):= `if`(n<1, 0, s(n-1)+ithprime(n)) end:
a:= n-> (t-> t(s(n))-t(s(n-1)))(i-> i*(i+1)/2):
seq(a(n), n=1..40);  # Alois P. Heinz, Mar 22 2023
  • Mathematica
    Module[{nn=50,pr},pr=Prime[Range[nn]];Total/@TakeList[Range[ Total[ pr]], pr]](* Requires Mathematica version 11 or later *) (* Harvey P. Dale, Oct 01 2017 *)
  • Python
    from itertools import islice
from sympy import nextprime
def A034956_gen(): # generator of terms
    a, p = 0, 2
    while True:
        yield p*((a<<1)+p+1)>>1
        a, p = a+p, nextprime(p)
A034956_list = list(islice(A034956_gen(),20)) # Chai Wah Wu, Mar 22 2023

Formula

From Hieronymus Fischer, Sep 27 2012: (Start)
a(n) = Sum_{k=A007504(n-1)+1..A007504(n)} k, n > 1.
a(n) = (A007504(n) - A007504(n-1))*(A007504(n) + A007504(n-1) + 1)/2, n > 1.
a(n) = (A000217(A007504(n)) - A000217(A007504(n-1))), n > 0.
If we define A007504(0) := 0, then the formulas above are also true for n=1.
a(n) = (A034960(n) + A000040(n))/2.
a(n) = A034957(n) + A000040(n). (End)

A034958 Divide primes into groups with prime(n) elements and add together.

Original entry on oeis.org

5, 23, 101, 311, 931, 1895, 3875, 6349, 10643, 18335, 25873, 39593, 55607, 71301, 94559, 127315, 167495, 204063, 258283, 315087, 369749, 451635, 533015, 640097, 779283, 902789, 1013795, 1159073, 1295871, 1457935, 1786691, 2002645, 2272221
Offset: 1

Views

Author

Patrick De Geest, Oct 15 1998

Keywords

Examples

			a(1) = 5 because the first 2 primes are 2 and 3 and 2 + 3 = 5.
a(2) = 23 because the next 3 primes are 5, 7, 11, and they add up to 23.
a(3) = 101 because the next 5 primes are 13, 17, 19, 23, 29 which add up to 101.
a(4) = 311 because the next 7 primes are 31, 37, 41, 43, 47, 53, 59 and they add up to 311.

Links

Crossrefs

Cf. A006003, A027441, A034956.
Cf. A007504, A046992, A034959, A034960, A180302.

Programs

  • Mathematica
    Join[{5},Total[Prime[Range[#[[1]]+1,#[[2]]]]]&/@Partition[ Accumulate[ Prime[ Range[40]]],2,1]] (* Harvey P. Dale, Oct 03 2013 *)
Module[{nn=33},Total/@TakeList[Prime[Range[Total[Prime[Range[nn]]]]], Prime[ Range[ nn]]]] (* Requires Mathematica version 11 or later *) (* Harvey P. Dale, Mar 16 2018 *)
s = 0; Total[Table[s = s + 1; Prime[s], {j, 33}, {n, Prime[j]}], {2}] (* Horst H. Manninger, Jan 17 2019 *)
  • PARI
    s(n) = sum(k=1, n, prime(k)); \\ A007504
a(n) = s(s(n)) - s(s(n-1)); \\ Michel Marcus, Oct 12 2018

Formula

From Hieronymus Fischer, Sep 26 2012: (Start)
a(n) = Sum_{k=A007504(n-1)+1..A007504(n)} A000040(k), n > 1.
a(n) = A007504(A007504(n)) - A007504(A007504(n-1)), n > 1.
If we define A007504(0) := 0, then the formulas are also true for n = 1.
(End)

A034960 Divide odd numbers into groups with prime(n) elements and add together.

Original entry on oeis.org

4, 21, 75, 189, 495, 897, 1683, 2565, 4071, 6641, 8959, 13209, 17835, 22317, 28623, 37577, 48439, 57401, 71623, 85697, 98623, 118737, 138195, 163493, 196231, 224321, 249775, 281945, 310759, 347249, 420751, 467801, 525943, 571985, 656047
Offset: 1

Views

Author

Patrick De Geest, Oct 15 1998

Keywords

Examples

			{1,3} #2 S=4;
{5,7,9} #3 S=21;
{11,13,15,17,19} #5 S=75;
{21,23,25,27,29,31,33} #7 S=189.

Links

Crossrefs

Cf. A006003, A027441, A034959.
Cf. A007504.
Cf. A000040, A034956, A034957, A034958, A046992, A061802.

Programs

  • Maple
    S:= n-> sum(ithprime(k), k=1..n): seq(S(n+1)^2-S(n)^2, n=0..40); # Gary Detlefs, Dec 20 2011
  • Mathematica
    Accumulate[Join[{2}, ListConvolve[{1, 1}, #]]]*# & [Prime[Range[50]]] (* Paolo Xausa, Jun 23 2025 *)
  • PARI
    a0(n) = vecsum(primes(n))^2 - vecsum(primes(n-1))^2; \\ Michel Marcus, Jun 16 2024
  • Python
    from itertools import islice
from sympy import nextprime
def A034960_gen(): # generator of terms
    a, p = 0, 2
    while True:
        yield p*((a<<1)+p)
        a, p = a+p, nextprime(p)
A034960_list = list(islice(A034960_gen(),20)) # Chai Wah Wu, Mar 22 2023

Formula

From Hieronymus Fischer, Sep 26 2012: (Start)
a(n) = Sum_{k=A007504(n-1)+1..A007504(n)} (2*k-1).
a(n) = A007504(n)^2 - A007504(n-1)^2.
a(n) = 2*A034957(n) + A000040(n).
a(n) = 2*A034956(n) - A000040(n).
a(n) = A034959(n) + A000040(n). (End)
a(n) = A061802(n)*A000040(n). - Marco Zárate, May 12 2023

A034957 Divide natural numbers in groups with prime(n) elements and add together.

Original entry on oeis.org

1, 9, 35, 91, 242, 442, 833, 1273, 2024, 3306, 4464, 6586, 8897, 11137, 14288, 18762, 24190, 28670, 35778, 42813, 49275, 59329, 69056, 81702, 98067, 112110, 124836, 140919, 155325, 173568, 210312, 233835, 262903, 285923, 327949, 355001, 393285
Offset: 1

Views

Author

Patrick De Geest, Oct 15 1998

Keywords

Comments

Natural numbers starting from 0,1,2,3,...

Examples

			{0,1} #2 S=1;
{2,3,4} #3 S=9;
{5,6,7,8,9} #5 S=35;
{10,11,12,13,14,15,16} #7 S=91.

Links

Crossrefs

Cf. A006003, A027441, A034956.
Cf. A046992, A034958, A034959, A034960.

Programs

  • Mathematica
    {1}~Join~Map[Abs@ Apply[Subtract, Map[PolygonalNumber, #]] &, Partition[Accumulate@ Prime@ Range@ 37 - 1, 2, 1]] (* Michael De Vlieger, Oct 06 2019 *)
Module[{nn=40,tprs},tprs=Total[Prime[Range[nn]]];Total/@TakeList[Range[0,tprs],Prime[Range[nn]]]] (* Harvey P. Dale, Apr 18 2025 *)
  • Python
    from itertools import islice
from sympy import nextprime
def A034957_gen(): # generator of terms
    a, p = 0, 2
    while True:
        yield p*((a<<1)+p-1)>>1
        a, p = a+p, nextprime(p)
A034957_list = list(islice(A034957_gen(),20)) # Chai Wah Wu, Mar 22 2023

Formula

From Hieronymus Fischer, Sep 27 2012: (Start)
a(n) = Sum_{k=A007504(n-1)+1..A007504(n)} (k-1), n > 1.
a(n) = (A007504(n) - A007504(n-1))*(A007504(n) + A007504(n-1) - 1)/2, n > 1.
a(n) = (A000217(A007504(n) - 1) - A000217(A007504(n-1) - 1)), n > 1.
If we define A007504(0):=0, then the formulas above are also true for n=1.
a(n) = A034959(n)/2.
a(n) = A034956(n) - A000040(n).
(End)
Showing 1-4 of 4 results.

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